冯·米塞斯分布的数学特性有哪些？

本文共计1124个文字，预计阅读时间需要5分钟。

von Mises Distribution（费·米塞斯分布）的随机模拟与参数估计笔记（二）

1. 参数估计计算子分析

在上一节中，我们讨论了von Mises Distribution的概率密度函数PDF和累积分布函数CDF。本节将深入探讨参数估计的计算子分析。

von Mises Distribution (冯·米赛斯分布)的随机模拟与参数估计的笔记（二） 1.参数估计算子分析

​ 在上一节中，我们讨论了von Mises Distribution的概率分布函数PDF和累计分布函数CDF，并给出了von Mises Distribution的随机模拟和参数估计matlab程序，其中在此我们就参数估计的细节进行补充。其基于最大似然参数估计算子，如下表：

​ 来源于《Statistical Distributions》

利用如下改进贝塞尔函数的关系求解参数\(\kappa\)，如下表达：

\[R=\frac{1}{n}\left[\left(\sum_{i=1}^{n} \cos x_{i}\right)^{2}+\left(\sum_{i=1}^{n} \sin x_{i}\right)^{2}\right]^{1 / 2} \]

\[\kappa \approx \begin{cases}2 R+R^{3}+\frac{5}{6} R^{5} \quad & R<0.53 \\ -0.4+1.39 R+\frac{0.43}{1-R} & 0.53 \leq R<0.85 \\ \frac{1}{R^{3}-4 R^{2}+3 R} & \text { other }\end{cases} \]

1.1 \(\mu\)参数估计分析matlab代码

function mu=circ_mean(alpha, w, dim) % % mu = circ_mean(alpha, w) % Computes the mean direction for circular data. % % Input: % alpha sample of angles in radians % [w weightings in case of binned angle data] % [dim compute along this dimension, default is 1] % % If dim argument is specified, all other optional arguments can be % left empty: circ_mean(alpha, [], dim) % % Output: % mu mean direction % % PHB 7/6/2008 % % References: % Statistical analysis of circular data, N. I. Fisher % Topics in circular statistics, S. R. Jammalamadaka et al. % Biostatistical Analysis, J. H. Zar % % Circular Statistics Toolbox for Matlab % By Philipp Berens, 2009 % berens@tuebingen.mpg.de - www.kyb.mpg.de/~berens/circStat.html if nargin < 3 dim = 1; end if nargin < 2 || isempty(w) % if no specific weighting has been specified % assume no binning has taken place w = ones(size(alpha)); else if size(w,2) ~= size(alpha,2) || size(w,1) ~= size(alpha,1) error('Input dimensions do not match'); end end % compute weighted sum of cos and sin of angles r = sum(w.*exp(1i*alpha),dim); % obtain mean by mu = angle(r); 1.2 \(\kappa\)参数估计的matlab代码

function kappa = circ_kappa(alpha,w) % % kappa = circ_kappa(alpha,[w]) % Computes an approximation to the ML estimate of the concentration % parameter kappa of the von Mises distribution. % % Input: % alpha angles in radians OR alpha is length resultant % [w number of incidences in case of binned angle data] % % Output: % kappa estimated value of kappa % % References: % Statistical analysis of circular data, Fisher, equation p. 88 % % Circular Statistics Toolbox for Matlab % By Philipp Berens, 2009 % berens@tuebingen.mpg.de - www.kyb.mpg.de/~berens/circStat.html alpha = alpha(:); if nargin<2 % if no specific weighting has been specified % assume no binning has taken place w = ones(size(alpha)); else if size(w,2) > size(w,1) w = w'; end end N = length(alpha); if N>1 R = circ_r(alpha,w); else R = alpha; end if R < 0.53 kappa = 2*R + R^3 + 5*R^5/6; elseif R>=0.53 && R<0.85 kappa = -.4 + 1.39*R + 0.43/(1-R); else kappa = 1/(R^3 - 4*R^2 + 3*R); end if N<15 && N>1 if kappa < 2 kappa = max(kappa-2*(N*kappa)^-1,0); else kappa = (N-1)^3*kappa/(N^3+N); end end

function r = circ_r(alpha, w, d, dim) % r = circ_r(alpha, w, d) % Computes mean resultant vector length for circular data. % % Input: % alpha sample of angles in radians % [w number of incidences in case of binned angle data] % [d spacing of bin centers for binned data, if supplied % correction factor is used to correct for bias in % estimation of r, in radians (!)] % [dim compute along this dimension, default is 1] % % If dim argument is specified, all other optional arguments can be % left empty: circ_r(alpha, [], [], dim) % % Output: % r mean resultant length % % PHB 7/6/2008 % % References: % Statistical analysis of circular data, N.I. Fisher % Topics in circular statistics, S.R. Jammalamadaka et al. % Biostatistical Analysis, J. H. Zar % % Circular Statistics Toolbox for Matlab % By Philipp Berens, 2009 % berens@tuebingen.mpg.de - www.kyb.mpg.de/~berens/circStat.html if nargin < 4 dim = 1; end if nargin < 2 || isempty(w) % if no specific weighting has been specified % assume no binning has taken place w = ones(size(alpha)); else if size(w,2) ~= size(alpha,2) || size(w,1) ~= size(alpha,1) error('Input dimensions do not match'); end end if nargin < 3 || isempty(d) % per default do not apply correct for binned data d = 0; end % compute weighted sum of cos and sin of angles r = sum(w.*exp(1i*alpha),dim); % obtain length r = abs(r)./sum(w,dim); % for data with known spacing, apply correction factor to correct for bias % in the estimation of r (see Zar, p. 601, equ. 26.16) if d ~= 0 c = d/2/sin(d/2); r = c*r; end 2 代码效果分析

clc clear all close all theta=pi/2; %设置模拟参数 kappa=50; n=3000; alpha=circ_vmrnd(theta,kappa,n); %生成制定参数的von-Mises分布的随机数 [thetahat1 kappa1]=circ_vmpar(alpha); %对其进行分布参数进行估计分析 %绘制模拟数据直方图 figure(1) hist(alpha,100); xlabel('Angle(弧度)'); ylabel('Frequency'); X = categorical({'Really value','Estimate value'}); %估计参数与模型参数对比 figure(2) subplot(1,2,1) bar(X,[theta,thetahat1]); ylabel('theta'); subplot(1,2,2) bar(X,[kappa,kappa1]); ylabel('kappa');